How to Graph Sine, Cosine, Tangent by Hand ✍
Plus their reciprocals: Cosecant, Secant, Cotangent Graphs
In our modern world, it may seem like graphing functions by hand is...well…a bit archaic. When you can easily enter any function you wish into Desmos or Wolfram Alpha from the comfort of your phone, why should you learn to graph functions longhand? Especially the more complicated functions like the ones we encounter in trigonometry?
Well besides the obvious likelihood that you are reading this because in some Trigonometry, Precalculus, or Calculus course your instructor is requiring you to graph sans technology, I’d like to present the argument that true understanding can only be gained through *doing*. The fact remains that you can’t learn math by standing on the sidelines. Math IS a participatory sport. You can read about math, listen to lectures, watch other people do math but if you don’t actually grab a pencil and paper and struggle through the problems, you will never learn mathematics.
In a day and age where we so easily misconstrue talking about doing things as actually doing things, math remains one of those disciplines that can’t be faked. If you are struggling to progress in math ask yourself if you are actually PRACTICING mathematics or are you thinking — or talking — about doing math instead?
Notice here how we say “practicing mathematics”, just like you practice an instrument or a sport, you are a beginner seeking improvement no matter what level of math you are currently at. Yes, you are going to get problems wrong. Yes, you are going to get confused. Yes, you are going to have questions and need help. That’s just part of the practice. 🙂
Alright, enough with the real talk. Let’s jump into sketching some trigonometric graphs!
What is a Parent Function?
You’ve probably heard the term Parent Function with relation to graphing. Parent functions are the OGs of functions. They are the unaltered forms of your equations. The archetypes. For example, the equation y=x is the parent linear function; the equation y=2x+1 is still a linear function but it is not the parent function. It is an altered form of the parent where we have changed the slope and y-intercept.
This idea of parent functions applies to all types of functions. In Trigonometry our main parent functions are: y=sin x, y=cos x, y=tan x, y=csc x, y=sec x, and y=cot x.
These are the functions you will be primarily working with in trigonometry (although as you progress you might learn about other interesting functions such the inverse or hyperbolic trig functions). So these are the functions we’ll be learning how to graph today!
The MVPs: Sine, Cosine, and Tangent
These three functions are definitely the all-stars of trig. You probably remember these functions from geometry when you were first introduced to right triangle trigonometry. Although the graphs of Sine, Cosine, and Tangent may seem completely different from what you learned in geometry, we are really just experiencing a new picture of the same geometric concept.
Sine, cosine, and tangent still represent SOH CAH TOA, but in trigonmetry, we learn about some new and creative ways we can represent this information.
Essentially what we have done here is scaled our special right triangles so that their hypotenuse is 1 unit in length, organized them around the origin of a xy-coordinate plane, and traded out the concept of degrees for radians (which is just angles measured in respect to a circular rotation).
When we lay out all these triangles around the origin, we can connect the outermost points to form a circle that has a radius of 1 unit. That circle is the Unit Circle and one of the most important topics in trig!
Now the points the lie on the circumference of the circle can be translated into the graphs we’ll be exploring today.
- According to SOH CAH TOA, we see that the x-coordinate relates to the cosine value (output) at the given angle rotation (input).
- The y-coordinate relates to the sine value (output) at the given angle rotation (input).
- And lastly tangent can be defined as cosine divided by sine, so the quotient y/x is the tangent value (output) at the given rotation (input).
Now we can graph a picture of this information that we gleaned from the Unit Circle, by creating a graph where the x-axis represents the rotation in radians, and the y-axis represents the value of our sine, cosine, or tangent functions at the given angle.
As I’m writing this, I’m realizing that this is a complicated set of relationships to understand, so I’m going to put together a separate tutorial to explain how these concepts all relate, but until then bear with me!
In the following video tutorial, I demonstrate how to graph the Sine, Cosine, and Tangent functions using a simple pattern and their periodic nature (ie repetitive) and then I relate it back to the Unit Circle, so you can see where those points come from 🙂
The Second String: Cosecant, Secant, Cotangent
The other three functions you will encounter in basic trigonometry are Cosecant, Secant, and Cotangent. These functions are often referred to as the reciprocals of Sine, Cosine, and Tangent because they are defined by the reciprocal (ie flipped upside down) ratios of SOH CAH TOA.
We can use these reciprocal identities to help us easily graph Csc, Sec, and Cot based on our knowledge of Sin, Cos, and Tan. Check it out ⬇
Advanced Topic: Trig Graph Transformations
So now that we have learned about the six standard parent functions of trigonometry, we are ready to take a look at how we can manipulate the parent functions to create and sketch interesting graphs by hand — which is one of the main reasons why we need to know how to graph the parent functions by hand!
Once you have a solid sense of how to graph your parent functions, you can apply some basic mathematical principles to yield the graphs for any variation of your parent functions 🙌
In the following tutorial, I explain how we can alter a parent function to produce x and y-axis reflections, horizontal and vertical stretches, compressions and shifts. Skip to 14 minutes in to see three examples of graphing transformed Tangent, Cosine, and Sine graphs.
Need More Math Help?🙏
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Thanks for joining me!